how much time will be needed for $32,000 to grow $34,299.49 if deposited at 7% compounded quarterly

Question

my calculator keeps giving me the answer in scientific notation and that is no correct. i know how to do the problem but for some reason i cannot get the right answer.

Answer 1

Hi Courtney,

 

Let's begin by writing the Compound Interest formula and identifying the variables and their values as given in the problem:

    F = P[1 + (r/p)]^pt

          F (future value) = $34,299.49

          P (initial value) = $32,000

          r (annual interest rate) = 7% = 7/100 = 0.07

          p (compound periods) = 4 (quarterly)

          t (time in years) = unknown

 

Next, let's solve the formula for time (t):

    34,299.49 = 32,000[1 + (0.07/4)]^4t

    34,299.49/32,000 = (1 + 0.0175)^4t

    1.071859063 = (1.0175)^4t

    log(1.0175)^4t = log(1.071859063)

    4t[log(1.0175)] = log(1.071859063)

    4t = log(1.071859063)/log(1.0175)

    4t = 0.0301376844/0.00753441789

    4t = 4.000001704

    t = 4.000001704/4

    t = 1.000000426

    t ≈ 1 year

 

We can see that our answer is 1 year with an extremely small remainder.  We can verify our answer of 1 year by plugging it back into our equation and checking that our answer brings us very close to the exact amount:

    34,299.49 ≅ 32,000[1 + (0.07/4)]^4t

    34,299.49 ≈ 32,000[1 + (0.07/4)]⁴⁽¹⁾

    34,299.49 ≈ 32,000(1 + 0.0175)⁴

    34,299.49 ≅ 32,000(1.0175)⁴

    34,299.49 ≅ 32,000(1.071859031)

    34,299.49 ≅ 34,299.489

 

We can see that plugging in our answer of 1 year brings us to within less than a penny of the exact amount.  In fact, rounding up to the nearest penny does indeed give us the exact amount !!

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan (Isaiah 53)